A new class of Volterra-type integral equations from relativistic quantum physics

Abstract

Here we study a new kind of linear integral equations for a relativistic quantummechanical two-particle wave function ψ(x 1 , x 2 ), where x 1 , x 2 are spacetime points. In the case of retarded interaction, these integral equations are of Volterra-type in the in the time variables, i.e., they involve a time integration from 0 to t i = x 0 i , i = 1, 2. They are interesting not only in view of their applications in physics, but also because of the following mathematical features: (a) time and space variables are more interrelated than in normal time-dependent problems, (b) the integral kernels are singular, and the structure of these singularities is non-trivial, (c) they feature time delay. We formulate a number of examples of such equations for scalar wave functions and prove existence and uniqueness of solutions for them. We also point out open mathematical problems.

Interacting relativistic quantum dynamics of two particles on spacetimes with a Big Bang singularity

“…In Sec. 4.1, we show that the theorems in [18] carry over to the case of flat FLRW universes. The main result are rigorous existence and uniqueness theorems for a class of integral equations with sufficiently regular interaction kernels for d = 1, 2, 3 dimensions (Thms.…”

“…In this section, we formulate the integral equation (2) on a (1+d)-dimensional Minkowski half-space for d = 1, 2, 3. After that, we state the existence and uniqueness results of the previous paper [18]. We shall give the full details, as they are necessary later in the paper.…”

“…The reasons for focusing on theses cases are the following. First and foremost, these are cases in which the time interval of integration becomes finite (see [18,Sec. 2.2] for a detailed illustration of the problems that can arise with infinite intervals).…”

“…In Sec. 2, we review the explicit form of the integral equation (2) on a Minkowski half-space with 1+d dimensions for d = 1, 2, 3 and summarize the results of the previous paper [18]. (The details will be needed later.)…”

Multi-time dynamics of the Dirac-Fock-Podolsky model of QED

“…In recent years, there has been a renewed interest in constructing mathematically rigoros multi-time models, see [5] for an overview. Some of the current efforts to understand Dirac's multi-time models focus on the well-posedness of the corresponding initial value problems [6,7,8,9,10], other works also ask the question how the multi-time formalism could be exploited to avoid the infamous ultraviolet divergence of relativistic QFT and how a varying number of particles by means of creation and annihilation processes can be addressed [11,12,13]. Beside being candidate models for fundamental formulations of relativistic wave mechanics, a better mathematical understanding of such multi-time evolutions may also be beneficial regarding more technical discussions, such as the control of scattering estimates on vacuum expectation values of products of interacting field operators; see e.g.…”